For the half quadratic models, let's begin with the following truncated quadratic regularization with gradient operator \(\nabla :=[\nabla _1,\nabla _2]\)  for image restoration [1]}, [2]}, [3]}, [4]}
\(&\inf _{u \in X} F(u ) =D(u) + \int _{\Omega }P^I(u )\mathrm {d}\sigma ,\quad P^I(u ):=\frac{\mu }{2}\min (|\nabla u |^2,\frac{\lambda }{\mu }),  \\&\inf _{u \in X} F(u ) = D(u) + \int _{\Omega }P^A(u)\mathrm {d}\sigma , \quad P^A(u):=\frac{\mu }{2}\sum _{i=1}^2\min (|\nabla _i u |^2, \frac{\lambda }{\mu }), \) 
